Introducing a new exchange functional by altering the electron density’s ionization dependency in density functional theory

Each of the exchange–correlation functionals in the density functional theory has been customized to particular systems or elements and has unique advantages and disadvantages. In one of the most recent research on exchange–correlation functionals, Chachiyo et al. present a relationship for exchange–correlation functional by assuming the simplest form of electron density. Their utilized electron density causes a systematic inaccuracy in the energy of the molecules since it does not fully account for the variation of the ionization energy for different atoms. We offer a novel relationship for exchange functional that improves the precision of the energy calculations for molecules by using the basic assumptions of the Chachiyo approach and correcting the electron density. Our density is directly related to the atom’s ionization energy. Our suggested functional was implemented for 56 molecules composed of atoms from the first, second, and third rows of the periodic table using Siam Quantum package. We discussed about the role of our functional on the reducing the computation error of dipole moment along with total, bonding and zero point energies. We also increased the portion of core electrons to improve the accuracy of the results.

Introducing a new exchange functional by altering the electron density's ionization dependency in density functional theory E. Rahmatpour & A. Esmaeili * Each of the exchange-correlation functionals in the density functional theory has been customized to particular systems or elements and has unique advantages and disadvantages.In one of the most recent research on exchange-correlation functionals, Chachiyo et al. present a relationship for exchange-correlation functional by assuming the simplest form of electron density.Their utilized electron density causes a systematic inaccuracy in the energy of the molecules since it does not fully account for the variation of the ionization energy for different atoms.We offer a novel relationship for exchange functional that improves the precision of the energy calculations for molecules by using the basic assumptions of the Chachiyo approach and correcting the electron density.Our density is directly related to the atom's ionization energy.Our suggested functional was implemented for 56 molecules composed of atoms from the first, second, and third rows of the periodic table using Siam Quantum package.We discussed about the role of our functional on the reducing the computation error of dipole moment along with total, bonding and zero point energies.We also increased the portion of core electrons to improve the accuracy of the results.
Quantum Monte Carlo (QMC) is employed to obtain one of the most precise estimates for the total energy of atoms in the framework of many particle systems 1 .As shown in Fig. 1, the mean absolute error (MAE) of QMC for molecules made from the first row of the periodic table and other rows, respectively, is 13.5 and 23.7 kcal/mol.Since QMC computations are very time-consuming, other approaches are currently being developed.Density functional theory (DFT) is one of them.It uses the electron correlation, kinetic and exchange energies, nuclearelectron interaction, and classical electron-electron Coulomb repulsion to determine the energy of a system.Since the correlation term is not taken into consideration in Hartree-Fock (HF) theory, the exchange energy determined precisely in HF cannot be employed in DFT 2 .Exchange energy can be calculated using a variety of approaches.These approaches depend on the electron density (n) as well as its gradients.Determining an exact exchange functional specially in the intermediate region, where electron density decays outside molecules between slow and rapid variation limit, is still under discussion.The primary aim of this work is to obtain exchange energy values that lead to results that are more accurate than QMC in all regions.
As predicted by the Thomas-Fermi model, the exchange energy enhancement in a system with slowly varying density is proportional to (1 + µS 2 ) , where S is a function of the reduced density gradient and is equal to According to the Kleinman model and taking into account an alternative external potential with a long wavelength as a perturbation, the µ coefficient has a constant value of 8/21 3 .In order to achieve more precise results for the total energy, Chachiyo et al. added the Bragg scattering condition for Fermi electrons and obtained a greater value for µ 1 .Then, using the assumption that the electron density decays expo- nentially as n(r) → Ne −ar , they offered a novel relationship for the exchange energy that led to accurate results for the total energy, particularly for a system with intermediate varying density 1 .As can be seen in Fig. 1, the error value of Chachiyo model (CM) for computing the total energy of molecules is acceptable and better than QMC.
The work by Chachiyo et al. actually misses a key aspect because the ionization energy is different for various atoms and the same relation for electron density should not be used for all of them.In this work, we utilized

Theoretical method
An interacting many-particle system is described by the DFT exclusively in terms of its density and the system energy is stated as [6][7][8] : F and V ion , respectively, stand for universal function and external potential in Eq. ( 2).The charge density contains all the information related to the system's ground state 9 .
And the density is calculated using the variational method as follows: Equation (2) may also be rewritten as follows: where E 0 xc [n] is the exchange correlation (XC) energy, which is defined by the following relationship: where ε xc n − → r represent exchange energy per electron.Gradient corrections are required in Eq. ( 6) to take into account the long-range gradient effects.The generalized gradient approximation (GGA) is the model that accomplishes this as: As mentioned earlier, the density in Eq. ( 7) is employed as follows in our model 4 : − 1 .For this density, |∇n(r)| is obtained as: (1) www.nature.com/scientificreports/In Eq. ( 8), I stands for the ionization energy and A is a constant.Equations ( 8) and ( 9) result: For a homogeneous electron gas, the Dirac exchange energy per electron, or ε x , equals 10 : The density determined by Eq. ( 8) yields the value of ε x as follows: Additionally, by using following equation for S 10 : As the reduced gradient parameter in the primary E x equation, the exchange energy will take the form: Therefore, S can be expressed as follows using Eq.(10): In the limit of r → ∞, Now, using Eq. ( 12) and this equation for S, we can derive the following equation for ε x : At the asymptotic limit of S → ∞ , the exchange energy density should behave as 10 : Since the value of the optimum coefficient should be equal to 1 at S = 0 limit, F x (S) is expressed as follows: The relationship between r and S can be identified through Eq. ( 16).Assuming: And using W as the Lambert function, we obtain: In this case, F x (S) is resulted as: ( . Equation ( 22) is obtained in the asymptotic range S → ∞.A weight function is required to extend the application of this formula to the intermediate range between S → ∞ and S → 0 limits.F x (S) should not diverge to infinity in the limit s → 0, thus we modify its functional as follows: The weight function is then introduced as ω(S) = 1 dS+1 where d is a constant that controls the weight function's speed from the ranges between low variation range to the asymptotic limit.Since the value of the functional at S = 0 must be equal to 1, we rewrite the weight function as follows: We derive the following series for the functional by using Taylor expansion up to the second order around the zero point: which is comparable to the reported ( 1 + µS 2 ) behavior for the low variation range 3 .
We obtain the following value for d in Eq. ( 25) by applying Chachiyo et al. 's suggested µ = 8/27: We ultimately achieved the non-experimental exchange functional in this work by using the value of d found in Eq. ( 26) and defining the new variable x = 4πS/3: The exchange energy was calculated using this F(x) as follows: � . (28) Vol.:(0123456789) We examine the accuracy of the introduced exchange-correlation energy for the set of atoms and molecules using Siam Quantum software and calculating Lambert function (Appendix A) 11,12 .

Results and discussion
The suggested exchange energy and functional in Eqs. ( 28) and (29) offers a very accurate total energy for a set of the atoms and molecules in the first and second rows of the periodic table.Figure 2 illustrates the errors of the estimated total energies of single atoms.For atoms smaller than Ne, the fit performance is excellent; however, it is less precise for bigger atoms.
A quick correction approach utilized to increase the accuracy of total energy 1 .This approach is predicated on taking into account how core electrons affect the quantity of energy.Exchange functional performs exceptionally well for atoms smaller than Ne, but it performs less accurately for larger atoms.A precise and straightforward adjustment for atoms and molecules can be applied to overcome this issue.For each core ion, such as the magnesium ion, we first compute the exchange energy error using the Hartree-Fock computations.Hartree-Fock orbitals are the results of this.Next, we compute the DFT exchange energy (28) using density equation and the Hartree-Fock exchange energy utilizing these orbitals.As shown in Fig. 2, the errors were decreased after applying the correction.The MAE of the entire set of 56 molecules was only 3.7 kcal/mol.(E xDFT -E HF ) for core electrons are nearly identical to the errors of the predicted total energies, as Fig. 3 illustrates.We take into account the 1S 2 orbital as the core electrons from the H atom to the Al atom in our error correction technique, and the 2S 2 2P 6 orbitals for the atoms larger than the Al atom (Fig. 3). (29) The mean absolute errors of the estimated total energies of single atoms.For atoms smaller than Ne, the fit performance is excellent; however, it is less precise for bigger atoms.In Fig. 4, this correction method depicted for molecules containing atoms from the first, second, and third rows of the periodic table.
For molecules containing atoms from the third row, this method very precisely illustrates the equality of the core electrons with the errors of predicted total energies.
The mean error (ME) of the total energy for the molecules containing the first, the second and the third rows of periodic table atoms is compared in Fig. 5.As shown in the caption of Fig. 5, the absolute mean error (MAE) for the first and second row molecules is better than the third row.These values for MAE are four times more accurate than the QMC result.
Tables 1 and 2 present a summary of the total energy produced by our developed XC functional as well as reference energies of atoms and molecules for computing the error value 13 .
The ME and MAE for neutral atoms obtained with our functional are less than CM, as Table 3 illustrates.Our exchange functional's ME is nearly zero for the total energy of the 56 molecules after rapplying correction, making it more accurate than the CM's reported 1 kcal/mol value, even though it yielded a 3.9 kcal/mol MAE, which is higher than the CM's reported 3.5 kcal/mol value.Table 4 illustrates the errors of the total energy for our model and the CM.
Additionally, we compared our total energy's ME with the results of the well-known B3LYP, BLYP, PBE, OLYP, QMC, and CE methods 1, [14][15][16] .The ME for the reference energies and for the molecules of the first and second rows of the periodic table is approximately 0.4 kcal/mol and which is same as our functional result as illustrated in Fig. 6 12 .Our functional estimates 17.3 kcal/mol MAE for the total energy without applying any corrections.For the CM, this error was equal to 18.5 kcal/mol.
Although the errors are only a few kcal/mol for molecules made up of atoms from the first and second rows of the periodic table, they drastically increase for molecules composed of toms from the third row.This increase in errors is caused by the core electrons.
Additionally, we used both corrected and uncorrected energies to evaluate molecules bond energy (E b ) using 1 : (30) Table 1.The total energy of examined atoms (versus Hartree) 13 .where E (A) and E (M) represent the total energies of atoms and molecules, respectively.In Fig. 7, the bond energy errors are displayed.MAE for molecules having atoms from the first and second rows of periodic table, from the third row of periodic table, and for all 58 molecules is equal to 4.6, 5.7 and 5.1, respectively.The MAE value for all molecules is decreased to 0.0 kcal/mol considering the core correction in Eq. (30).In this case, the atomic energies from reference 13 , along with the corrected molecule energy are employed.In Fig. 8, the MAE for our bond energy is compared with the QMC and CM approaches.Table 5 compares the bond energy errors for our model and the CM.Dipole moments are also analyzed using our employed electron density at this work.Our results and experimental data for dipole moments are summarized in Table 6.

Atoms
As summarized in Table 7, MAE is obtained 0.09 Debye which is comparable to the DFT-based estimations and better than CM 17 .
In contrast to classical mechanics, quantum systems fluctuate around zero-point energy (E ZP ) even at absolute zero temperature.In order to calculate vibration frequencies, the accuracy of E ZP should be increased.E ZP can be calculated using Eq.(31) as 18 : where i represent the frequency of a certain molecule and h is the Planck constant.Table 8 provides an overview of our findings and the experimental data for E ZP .
The MAE of the E ZP , as shown in Table 8, is 0.11 kcal/mol, which is comparable to the Chachiyo prediction 1 .In this investigation, the QZP-g basis set was employed.
We compared the errors of zero point energy for our and CM in Table 8.As a result, in addition to the total energies, our exchange correlation can also reliably predict the dipole moment, E B and ZP .
We curious to see our functional performed with various organic and solid substances.We also achieved satisfactory results for their dipole moment, zero-point energy, and total energy, as Table 9 shows.Siam-Quantum can be used to compute additional variables, such as molecular vibrations and the related normal modes and forces, which were computed for solids.Table 10 presents an overview of the results achieved for these parameters.

Conclusion
In this paper, we derive an accurate and straightforward exchange functional that can be applied in the intermediate, slow and fast density variation limits.According to calculations on first, second and third rows of periodic table neutral atoms and 56 molecules, our exchange functional accurately predicts the total energy, dipole moment, bond and zero point energies.The total energy mean error of our functional is 0.0 for examined 56 molecules which shows that it concludes more accurate results than other exchange functionals.The mean absolute error of the total energy for the mentioned molecules are calculated was obtained 3.9 kcal/mol which   www.nature.com/scientificreports/ is higher than the result of CM calculations but is better than the QMC result.The error of our functional for the third row of the periodic table's atoms was greater than that of the first and second rows' atoms because of the role of core atoms.Our functional results for neutral atoms are comparable to CM as evidenced by its low ME and MAE.The MAE for the estimated zero-point energy and dipole moments also confirms the accuracy of our new exchange functional.

Figure 3 .
Figure 3. (Ex-DFT-EHF) core correction and the error of total energy for neutral atoms.

Figure 4 .Figure 5 .
Figure 4. (Ex-DFT-EHF) core correction and the error of total energy for molecules including (A) first and second row atoms and (B) third row atoms.

Figure 7 .
Figure 7.The bond energies mean absolute error (kcal/mol) obtained by our new compared to the experimental value (A) molecules containing atoms of the first and second rows of the periodic table and (B) molecules containing atoms of the third row of the periodic table.

Figure 8 .
Figure 8.The bond energies mean absolute error (kcal/mol) obtained by various models the experimental for (A) molecules containing atoms of the first and second rows of the periodic table and (B) molecules containing atoms of the third row of the periodic table.
Total energy prediction error in terms of kilojoules/mol obtained from quantum Monte Carlo method and Chachiyo model for molecules containing atoms of the first and second rows of the periodic table 1 .

Table 3 .
The comparison of the our and Chachiyo models total energy errors for neutral atoms.The unit of every energy is kcal/mol. 1.

Table 4 .
The comparison of the our and Chachiyo models total energy errors for examined 56 molecules.The unit of every energy is kcal/mol. 1.
Figure6.The total mean absolute error (kcal/mol) of DFT approaches compared the experimental for that contain (A) atoms from the first and second rows of the periodic table and (B) atoms from the third row of the periodic table.

Table 5 .
The comparison of the bond energy errors for our and Chachiyo models.The unit of every energy is kcal/mol.

Table 6 .
The experimental and our results for zero point energy and dipole moment of molecules.The units are in Debye 5 .

Table 7 .
The comparison of the dipole moment errors for our and Chachiyo models.The unit of every energy is kcal/mol. 1.

Table 8 .
The comparison of the zero point energy errors for our and Chachiyo models.The unit of every energy is kcal/mol. 1.

Table 10 .
Molecular vibration properties of some solid materials' .